What happens if you plug it in while it is is off and then try to turn it on?
Also, while you think the battery may have enough power to keep running it, it is possible that the battery is not holding as much power as it used to. Which is normal for a battery. Over time they hold less and less, but can still say they have more of a % of a charge than they do.
If that is the case, I would replace the battery.
If, however, it won't turn on at all, after turning off, until the battery is charged, and then will turn on once you unplug the charger, then I would replace the charger as it is starting to go bad.